When a system is swept through a quantum critical point, the quantum Kibble-Zurek mechanism makes universal predictions for quantities such as the number and energy of excitations produced. This mechanism is now being used to obtain critical exponents on emerging quantum computers and emulators, which in some cases can be compared to matrix product state (MPS) numerical studies. However, the mechanism is modified when the divergence of entanglement entropy required for a faithful description of many quantum critical points is not fully captured by the experiment or classical calculation. In this Letter, we study how low-energy dynamics of quantum systems near criticality are modified by finite entanglement, using conformally invariant critical points described approximately by a MPS as an example. We derive that the effect of finite entanglement on a Kibble-Zurek process is captured by a dimensionless scaling function of the ratio of two length scales, one determined dynamically and one by the entanglement restriction. Numerically we confirm first that dynamics at finite bond dimension χ is independent of the algorithm chosen, then obtain scaling collapses for sweeps in the transverse field Ising model and the three-state Potts model. Our result establishes the precise role played by entanglement in time-dependent critical phenomena and has direct implications for quantum state preparation and classical simulation of quantum states.
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